2024/10/31 by Kang, Mohammad Behzad, Salch, Andrew · 1 citation
#14D05 #20J06 #55Q10 #55Q45 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.24171
We calculate the cohomology of the extended Morava stabilizer group of height n, with trivial mod p coefficients, for all heights n and all primes p>>n. The result is an exterior algebra on n generators. A brief sketch of the method: we introduce a family of deformations of Ravenel's Lie algebra model L(n,n) for the Morava stabilizer group scheme. This yields a family of DGAs, parameterized over an affine line and smooth except at a single point. The singular fiber is the Chevalley-Eilenberg DGA of Ravenel's Lie algebra. Consequently the cohomology of the singular fiber is the cohomology of the Morava stabilizer group, at large primes. We prove a derived version of the invariant cycles theorem from Hodge theory, which allows us to compare the cohomology of the singular fiber to the fixed-points of the Picard-Lefschetz (monodromy) operator on the cohomology of a smooth fiber. Finally, we use some new methods for constructing small models for cohomology of reductive Lie algebras to show that the cohomology of the Picard-Lefschetz fixed-points on a smooth fiber agrees with the singular cohomology H^*(U(n);\mathbbFp) of the unitary group, which is the desired exterior algebra.